derives the distribution of Y^2 (the notation is different of course).
Y^2 follows a chi-square distribution of degree 1. You can either look up the moment generating function or derive it from the definition.
Standard theorem: Var(aZ + b) = a^2 Var(Z). So Var(2Y^2 + 3) = 4 Var(Y^2).
To get Var(Y^2) you can either look it up or derive it from E[(Y^2)^2] - (E[Y^2])^2. The required expectations can be got from the moment generating function of Y^2.